Parameterised moduli spaces of surfaces as infinite loop spaces

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• Andrea Bianchi (Copenhagen)
• Wednesday 30 November 2022, 16:00-17:00
• MR13.

If you have a question about this talk, please contact Oscar Randal-Williams.

We consider the E_2-algebra consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of its group completion: it is the product of \Omega\infty MTSO with the free \Omega\infty-space generated by a certain space X. This extends the classical result due to Madsen and Weiss to the setting of surface bundles over S^1.

The proof consists of two inputs. The first input, on which my talk will focus, is an analysis of centralisers of mapping classes in generic mapping class groups Gamma_{g,n}, for g>=0 and any n>=1: this uses standard techniques of the theory of mapping class groups, such as arc complexes. The other input, which I will only briefly mention, is a generalised theory of operads with homological stability in the setting of coloured operads.

This is joint work with Florian Kranhold and Jens Reinhold.

This talk is part of the Differential Geometry and Topology Seminar series.

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